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Z k -actions fixing point ∪ Vⁿ

Pedro L. Q. Pergher — 2002

Fundamenta Mathematicae

We describe the equivariant cobordism classification of smooth actions ( M m , Φ ) of the group G = Z k on closed smooth m-dimensional manifolds M m for which the fixed point set of the action is the union F = p ∪ Vⁿ, where p is a point and Vⁿ is a connected manifold of dimension n with n > 0. The description is given in terms of the set of equivariant cobordism classes of involutions fixing p ∪ Vⁿ. This generalizes a lot of previously obtained particular cases of the above question; additionally, the result yields...

Commuting involutions whose fixed point set consists of two special components

Pedro L. Q. PergherRogério de Oliveira — 2008

Fundamenta Mathematicae

Let Fⁿ be a connected, smooth and closed n-dimensional manifold. We call Fⁿ a manifold with property when it has the following property: if N m is any smooth closed m-dimensional manifold with m > n and T : N m N m is a smooth involution whose fixed point set is Fⁿ, then m = 2n. Examples of manifolds with this property are: the real, complex and quaternionic even-dimensional projective spaces R P 2 n , C P 2 n and H P 2 n , and the connected sum of R P 2 n and any number of copies of Sⁿ × Sⁿ, where Sⁿ is the n-sphere and n is not...

Z k -actions with a special fixed point set

Pedro L. Q. PergherRogério de Oliveira — 2005

Fundamenta Mathematicae

Let Fⁿ be a connected, smooth and closed n-dimensional manifold satisfying the following property: if N m is any smooth and closed m-dimensional manifold with m > n and T : N m N m is a smooth involution whose fixed point set is Fⁿ, then m = 2n. We describe the equivariant cobordism classification of smooth actions ( M m ; Φ ) of the group G = Z k on closed smooth m-dimensional manifolds M m for which the fixed point set of the action is a submanifold Fⁿ with the above property. This generalizes a result of F. L. Capobianco,...

On the Extension of Certain Maps with Values in Spheres

Carlos BiasiAlice K. M. LibardiPedro L. Q. PergherStanisław Spież — 2008

Bulletin of the Polish Academy of Sciences. Mathematics

Let E be an oriented, smooth and closed m-dimensional manifold with m ≥ 2 and V ⊂ E an oriented, connected, smooth and closed (m-2)-dimensional submanifold which is homologous to zero in E. Let S n - 2 S be the standard inclusion, where Sⁿ is the n-sphere and n ≥ 3. We prove the following extension result: if h : V S n - 2 is a smooth map, then h extends to a smooth map g: E → Sⁿ transverse to S n - 2 and with g - 1 ( S n - 2 ) = V . Using this result, we give a new and simpler proof of a theorem of Carlos Biasi related to the ambiental bordism...

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